Current Issues in Statistical Shape Analysis

The Leeds Conference "Current Issues in Statistical Shape Analysis" from
April 5-7, 1995 was a great success.

Held in Fairbairn house - a complete facility for housing, conference,
breakfast and lunch. One didn't have to stir far to be comfortably involved
in the conference. The Leeds staff and especially Kanti Mardia and C. A.
Gill made it a wonderful exciting experience for all fortunate enough to
attend. In his welcoming talk Kanti set the tone of the entire 3 days by
invoking the Jain idea of ANEKANTVAD - freely translated as "Striving for
Balance". That was accomplished as much as at any conference I have attended
where there was a mix of professions, viewpoints, and attitudes.
Among the 80+ attendees were 14 each from the USA, Canada, and non UK
Europe, a sizable contingent from the UK and a few others from as far away
as Australia.

The Opening Address "Looking at Geodesics" by Professor David Kendall was on
the 5th of April. Then followed sessions on Procrustes, Shape Geometry,
Image Analysis and Computer Vision. The complete program was given
earlier on MORPHMET and is in the archive there.
A panel discussion was the last event of the conference in which several
issues raised at the conference were discussed.
Two poster sessions gave an opportunity for additional presentations, and
especially applications of morphometrics to a variety of data sets in biology,
anthropology, and paleontology.
The Proceedings were available to all registered, and can be obtained as
detailed at the end of this review.
I borrowed heavily from the presentations and printed papers of Kanti Mardia,
John Kent and especially that of Ian Dryden as they presented fine previews,
summaries, and a review of accomplishments.
D. G. Kendall's opening address gave some insight to looking at
projections of shape space for 4 landmarks in 3 space - all tetrahedra. This
shape space has 5 dimensions.

Session I. Procrustes and Mathematical Statistics Issues

Gower offered some new thoughts on distances presented in Euclidean
Distance Matrices for landmarks. Goodall updated his 1991 paper with
additional points on consistency, likelihood and estimation of covariance
parameters. He described Euclidean Shape Tensor Analysis (ESTA)- a
method based on subsets of landmarks - Euclidean Distance Matrix Analysis is
a special case. A poster presentation of ESTA provided additional material
and an example using data on Apert's Syndrome in humans. Le considered a
special form of mean shape.
Dryden and co-workers considered a case of a series of triangles sharing
points over a set of objects in 2D. Some new results were obtained and
applied to regularity in human muscle fiber cross-sections. Lele and Cole
summarized EDMA and proposed a new version based on differences
(rather than ratios) of forms. They studied the power of a new test
using simulations.

Session II. Overview and Shape Geometry

Kanti Mardia gave one of his sterling rapid fire overviews of statistical shape
analysis - including developments, distributions, tangent plane approximations,
principal component analysis, Kriging with derivatives and also touched on
image analysis. Bhavnagi defined a Markov process to use the results to
classify objects as simple shapes in vision applications. Molchanov considered
shape analysis of more abstract sets than the usual ones containing equal
numbers of landmarks - a most intriguing development. Small and Lewis have
developed a landmark free method which superimposes pixel lattices of
objects. The method was applied to Iron Age broaches.

Session III. Image Analysis and Computer Vision

Markov Chain Monte Carlo methodology was discussed by Green - which can
be applied to situations where the dimension of the parameter space is
unknown - for example where an image contains an unknown number of
objects. Practical problems such as object recognition in Bayesian image
analysis can be addressed. A fun presentation was given by Marchant using
"snakes" for locating objects in images. A snake is a physical analog for
finding boundaries and compartments. "Sidewinders" also come into action.
Applications were for locating and determining location and shape of pigs
from dorsal photographs - an obvious economic application. Cootes
described object recognition using Active Shape Models. This is a principal
components approach to the decomposition of shape variability in a Procrustes
tangent space. New refinements were given.
The last three papers dealt with computer vision tasks - edge recognition,
tracking objects and depth recognition.

Session IV. Morphometrics and Shape Geometry

Bookstein gave the opening talk on the synthesis of multivariate analysis and
geometrical deformations. The use of relative warps was reviewed with two
examples - Foraminifera landmarks, and brain scan images which can be
analyzed to identify schizophrenics and normals. Rohlf summarized a
simulation study that showed that two group multivariate analysis of
variance using partial warp scores and multiple regression of uniformly
distributed random numbers on the scores in tangent space gave correct
significance levels under a variety of point configurations and covariance
structure for Gaussian data. Sampson looked at shape changes in
the heart left ventricle using Procrustes superimposition of points along the
outline. He was able to decompose the shape variation. Finally W. D. K.
Green gave an alternative construction of Kendall's shape space for triangles,
with a new proof and some incites into distribution results for triangle shapes.
Kent gave the concluding paper - summarizing current statistical approaches
to shape analysis, with outstanding issues, and pointers toward further
development - focusing on landmark-based methods. Consistency, tangent
spaces and different modeling strategies were considered. This paper best
summarized many of the current issues and will be discussed in greater length
than the others. Three situations of landmark distributions are considered by
Kent 1) the covariance matrix is such that the coordinates of all the landmarks
have identical variances (isotropy - no correlations); 2) their are structured
correlations between and within landmarks; 3) the most general case where
the covariance matrix is unrestricted. Kent has worked out some elegant
distributions for these situations, but they are not practical for applications.
For "concentrated data" (occupying a small part of shape space - it has been
suggested that at least in biological comparisons though shapes may look very
different to the biologist, they really only occupy a small part of all possible
shapes. A mammal skull is by and large the same over most mammals.), an
alternative model is to consider case 3) in the tangent space to shape space.
"It should be emphasized that for concentrated data any possible
inconsistencies will usually be swamped by the variability in the data. Thus
possible inconsistency of these methods is not usually an important statistical
problem".
He then went on to state the consistency properties of several of the popular
methods, but with this caveat in mind. The issue of consistency therefore no
longer seems so important.

Posters

The posters were divided into two sessions, and there was a 5 minute
presentation for each one by the authors. I will briefly summarize the posters
paralleling Session IV based on their abstracts. These were applications.
Corti et al. applied relative warp analysis to a super species of mole rats and
found shape differences associated with chromosome number, soil type and
locality. Dean et al. continue work on deformable templates - finding space
curves (ridge curves) of maximum curvature in order to tile the human skull
surface patches. These lead to a deficient coordinate notion. The templates
allow superimposition and precise topological reference to significant
features. Kucera looked at random walks of shape coordinates in lineages of
Foraminifera. Penin and Baylac have recorded 29 3D landmarks on 140 skulls
of great apes. Principal components of a Procrustes fit were used for further
analysis. Similar patterns of growth were observed, and a functional
descriptive framework elucidated. Wood and Wood are beginning an analysis
of articulating surfaces of the ankle using laser scanning in apes and man. The
goal is to compare fossil hominids and quantify differing locomotor regimes.
In an earlier session Renaud found it useful to study outlines of a lineage of
fossil rodent teeth using Zahn and Roskies "inverse of the curvature radius" to
avoid problems of standardization, and claimed interpretable phylogeny.

The high point of the meeting was supposed to be the Panel Discussion at the
end - however the biological audience was somewhat frustrated. A discussion
of efficiency of the methods by Bookstein and Lele considered different ideas
of efficiency, and therefore the discussion wasn't! Lele presented some
simulation results for one simple data set for a fixed number of landmarks;
while Bookstein discussed the affect of increasing numbers of landmarks.
However, much of the differences and a great deal of clarification has
appeared on MORPHMET since the meeting (see the archives for April
if you want to review the discussions)
The meeting reflected much of the state of shape analysis, and it is clear that
much more work remains to be done. I for one, would like to see a
continuation of the simulations begun by Rohlf, and extended to calculations
of power.
The data presentations were representative of what is being done with
landmark shape analysis, eg. in biology, and it is useful for statisticians
to see more of the kinds of biological data applications and the problems they
entail for the biologist.

Leslie F. Marcus,
Professor of Biology,
Queens College of CUNY.
Department of Invertebrates, American Museum of Natural History,
CPW at 79th, New York, NY 10024

HOW TO OBTAIN A COPY OF:
Proceedings in
CURRENT ISSUES IN STATISTICAL SHAPE ANALYSIS
International Conference,
held in Leeds, UK, 5-7 April 1995
Edited by K.V.Mardia, C.A.Gill
Department of Statistics,
University of Leeds, UK
Leeds University Press
ISBN 0 85316 161 5